It is well known that for any inner function θ defined in the unit disk D, the following two conditions: (i) there exists a sequence of polynomials {pn}n such that limn→∞θ(z)pn(z)=1 for all z∈D and (ii) supn∥θpn∥∞<∞, are incompatible, i.e., cannot be satisfied simultaneously. However, it is also known that if we relax the second condition to allow for arbitrarily slow growth of the sequence {θ(z)pn(z)}n as |z|→1, then condition (i) can be met for some singular inner function. We discuss certain consequences of this fact which are related to the rate of decay of Taylor coefficients and moduli of continuity of functions in model spaces Kθ. In particular, we establish a variant of a result of Khavinson and Dyakonov on nonexistence of functions with certain smoothness properties in Kθ, and we show that the classical Aleksandrov theorem on density of continuous functions in Kθ is essentially optimal. We consider also the same questions in the context of de Branges–Rovnyak spaces H(b) and show that the corresponding approximation result also is optimal.